Question

suppose a random variable, x, arises from a binomial experiment. suppose n = 6, and p = 0.19. write the probability distribution. round to six decimal places, if necessary. x p(x) 0 1 2 3 4 5 6 select the correct histogram.

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is $P(x)=C(n,x)\times p^{x}\times(1 - p)^{n - x}$, where $C(n,x)=\frac{n!}{x!(n - x)!}$, $n = 6$, $p=0.19$, and $1-p = 0.81$.

Step2: Calculate $P(0)$

$C(6,0)=\frac{6!}{0!(6 - 0)!}=1$, then $P(0)=C(6,0)\times(0.19)^{0}\times(0.81)^{6}=1\times1\times0.282429536481\approx0.282430$.

Step3: Calculate $P(1)$

$C(6,1)=\frac{6!}{1!(6 - 1)!}=\frac{6!}{1!5!}=6$, then $P(1)=C(6,1)\times(0.19)^{1}\times(0.81)^{5}=6\times0.19\times0.3486784401\approx0.394393$.

Step4: Calculate $P(2)$

$C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times1}=15$, then $P(2)=C(6,2)\times(0.19)^{2}\times(0.81)^{4}=15\times0.0361\times0.43046721\approx0.233688$.

Step5: Calculate $P(3)$

$C(6,3)=\frac{6!}{3!(6 - 3)!}=20$, then $P(3)=C(6,3)\times(0.19)^{3}\times(0.81)^{3}=20\times0.006859\times0.531441\approx0.073588$.

Step6: Calculate $P(4)$

$C(6,4)=\frac{6!}{4!(6 - 4)!}=15$, then $P(4)=C(6,4)\times(0.19)^{4}\times(0.81)^{2}=15\times0.00130321\times0.6561\approx0.012899$.

Step7: Calculate $P(5)$

$C(6,5)=\frac{6!}{5!(6 - 5)!}=6$, then $P(5)=C(6,5)\times(0.19)^{5}\times(0.81)^{1}=6\times0.0002476099\times0.81\approx0.001203$.

Step8: Calculate $P(6)$

$C(6,6)=\frac{6!}{6!(6 - 6)!}=1$, then $P(6)=C(6,6)\times(0.19)^{6}\times(0.81)^{0}=1\times0.000047045881\times1\approx0.000047$.

Answer:

x	P(x)
1	0.394393
2	0.233688
3	0.073588
4	0.012899
5	0.001203
6	0.000047